Given three parallel linesFind$ \alpha,\beta $αααβββ383838140

$ \alpha=38 $ $ \beta=140 $

$ \alpha=142 $ $ \beta=140 $

Find α and β: Parallel Lines with 38° and 140° Angles

Q: How do I know which angles are corresponding?

Corresponding angles are in the same position when a transversal crosses parallel lines. They're like mirror images - same spot on each parallel line!

Q: Why does α = 142° instead of 38°?

Because α and the 38° angle are supplementary (they form a straight line). Supplementary angles always add up to 180°, so $ \alpha = 180° - 38° = 142° $.

Q: What's the difference between corresponding and alternate angles?

Corresponding angles are in the same relative position on parallel lines. Alternate angles are on opposite sides of the transversal. Both types are equal when lines are parallel!

Q: How do I find β when I know the 140° angle?

The 140° angle and β are supplementary because they form a straight line. So $ \beta = 180° - 140° = 40° $.

Q: Can I solve this without knowing about parallel lines?

No! The parallel line property is essential here. Without parallel lines, we couldn't use the corresponding and supplementary angle relationships to find α and β.

Angle Relationships with Parallel Line Intersections

Given three parallel lines

Find $\alpha,\beta$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Calculate angle A,B

00:05 Parallel lines according to the given data, marked with letters

00:12 Corresponding angles are equal

00:17 Alternate angles sum to 180 between parallel lines

00:27 Let's isolate angle A

00:37 This is angle A

00:52 Lines are parallel according to the given data

00:57 Alternate angles are equal between parallel lines

01:07 Supplementary angle to 180

01:17 Let's isolate angle B

01:22 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Given three parallel lines

Find $\alpha,\beta$

Step-by-step solution

We will mark the angle opposite the vertex of 38 with the number 1, therefore, angle 1 is equal to 38 degrees.

We will mark the angle adjacent to angle $\beta$ with the number 2. And since angle 2 corresponds to the angle 140, angle 2 will be equal to 140 degrees

Since we know that angle 1 is equal to 38 degrees we can calculate the angle $\alpha$ $\alpha=180-38=142$

Now we can calculate the angle $\beta$

180 is equal to angle 2 plus the other angle $\beta$

Since we are given the size of angle 2, we replace the equation and calculate:

$\beta=180-140=40$

Final Answer

$\alpha=142$ $\beta=40$

Key Points to Remember

Essential concepts to master this topic

Parallel Lines: Corresponding angles are equal, supplementary angles sum to 180°
Technique: Use α = 180° - 38° = 142° for supplementary angles
Check: Verify α + 38° = 180° and β + 140° = 180° ✓

Common Mistakes

Avoid these frequent errors

Confusing corresponding angles with supplementary angles
Don't assume α = 38° just because they look related = wrong answer! Corresponding angles are only equal when they're in the same relative position. Always identify if angles are supplementary (add to 180°) or corresponding first.

Practice Quiz

Test your knowledge with interactive questions

It is possible for two adjacent angles to be right angles.

FAQ

Everything you need to know about this question

How do I know which angles are corresponding?

Corresponding angles are in the same position when a transversal crosses parallel lines. They're like mirror images - same spot on each parallel line!

Why does α = 142° instead of 38°?

Because α and the 38° angle are supplementary (they form a straight line). Supplementary angles always add up to 180°, so $\alpha = 180° - 38° = 142°$ .

What's the difference between corresponding and alternate angles?

Corresponding angles are in the same relative position on parallel lines. Alternate angles are on opposite sides of the transversal. Both types are equal when lines are parallel!

How do I find β when I know the 140° angle?

The 140° angle and β are supplementary because they form a straight line. So $\beta = 180° - 140° = 40°$ .

Can I solve this without knowing about parallel lines?

No! The parallel line property is essential here. Without parallel lines, we couldn't use the corresponding and supplementary angle relationships to find α and β.

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